Does History of Science Treat of the History of Science? The Case of Mathematics

For a good survey of cliometrics see Fogel R. W. , “‘Scientific’ history and traditional history”, in Logic, methodology and philosophy of science VI (Hannover 1979) (Amsterdam, 1982), 15–61.

This little known work is de B. Garçao-Stockler F. , Ensaio historico sobre a origiem e progresses das matematicas em Portugal (Paris [sic!], 1819); Saraiva L. (Lisbon) is studying its interesting author. Histories of various nations (for example, Italy), had already included discussions of pertinent developments in mathematics.

In a beautiful example of national characteristics, moves for an English edition were rejected out of hand in Britain, even though some British mathematicians had contributed to the German edition.

See Günther S. , “Geschichte der Mathematik”, in Cantor M. (ed.), Vorlesungen über Geschichte der Mathematik, iv (Leipzig, 1908), 1–36.

These figures include the remarkable George Sarton, who published two bibliographical guides on the history of mathematics (from which the quotation at the head of Section 2 below is taken) and on the history of science respectively: The study of the history of mathematics and The study of the history of science (both Cambridge, Mass., 1936; volumes reprinted as one, New York, 1957). His journals Isis and Osiris took papers on the history of mathematics from time to time. He even taught a course in the history of mathematics at Harvard University, according to Sayili A. , “George Sarton and the history of science”, Türk tarih kurumu belleten, xlvii (1983, pb. 1984), 499–525, p. 503.

See my “History of mathematics”, in Dorling A. (ed.), Use of mathematics literature (London, 1977), 60–77.

See May K. O. (ed.), Bibliography and research manual of the history of mathematics (Toronto, 1973); and Dauben J. W. (ed.), The history of mathematics from Antiquity to the present: A selective bibliography (New York, 1984).

See U. Bottazzini's contributions to Rossi P. (ed.), Storia della scienza moderna e contemporanea (5 vols, Turin, 1988), passim.

An excellent example of the increase in the recent quality of work on mathematics since 1800 is provided by comparing conference proceedings: Birkhoff G. D. (ed.), “Proceedings of the American Academy Workshop on the evolution of modern mathematics…”, Historia mathematica, ii (1975), 425–615 with Rowe D. and McCleary J. (eds), History of modern mathematics (2 vols, New York, 1989).

Sarton , Mathematics (ref. 5), 4,3.

As an example of the development of History of Mathematics relative to Histories of accepted sciences, comparison of the section of abstracts in Historia mathematica with the quarterly Current work in the history of medicine suggests that work in the latter area is the larger by between fifteen and twenty times.

See Kohlstedt S. G. and Rossiter M. W. (eds), Historical writing on American science (Philadelphia, 1985).

Rowe D. and Hunger K. Parshall are currently completing a volume on the history of American mathematics.

See Ogilvie M. B. , “Marital collaboration: An approach to science”, in Abir-Am P. G. and Outram D. (eds), Uneasy careers and intimate lives (New Brunswick and London, 1987), 104–25. On the Youngs, see principally my “A mathematical union: William Henry and Grace Chisholm Young”, Annals of science, xxix (1972), 105–86; their Nachlass is in the Archives of Liverpool University.

See Fauvel J. and Gray J. J. (eds), The history of mathematics: A reader (London, 1987).

See Kolata G. , “Math archive in disarray”, Science, ccx (1983), 940.

If this sounds to be an extreme construal, let me recall an occasion in 1970 when I was allowed to examine the (totally disordered) archives of the Institut Mittag-Leffler, a mathematics research institute in the suburbs of Sweden. In the course of finding materials relating to my own concerns, I came across mounds of completely unknown manuscripts for a major figure of the nineteenth century, and smaller collections for several other such figures (see my “Materials for the history of mathematics in the Institut Mittag-Leffler”, Isis, lxii (1971), 363–74), but was quite unable to arouse the interest of the director in them. By coincidence, just at that time the discovery of some rather trivial materials of Jane Austen was greatly exciting the literary and historical world.

The difference between these two questions is discussed in the general context of historiography in my “What do theories talk about? A critique of Popperian fallibilism, with especial reference to ontology”, Fundamenta scientiae, vii (1986), 177–221, esp. sect. 10. As an issue, it is underrated by historians and philosophers in general.

The extent to which mathematicians' conception of history affects their historical work is very profound: Here are two examples, involving eminent mathematicians.

Some of the individual articles in Dieudonné J. (ed.), Abrégé d'histoire des mathématiques 1700–1900 (2 vols, Paris, 1978) are very good; but the book is conceived to reflect a certain modernist conception of mathematics, so that about eighty per cent of the subject developed during the period in question is omitted entirely!! No articles are devoted to mechanics or the various branches of mathematical physics, although for most of the period they were the principal concern. Further, the differences between the two centuries arising from the vast increase in professionalization after 1800 are barely recorded. See my review in Annals of science, xxxvi (1979), 653–5.

Birkhoff G. D. (ed.), A source book in classical analysis (Cambridge, Mass., 1973) contains a (sometimes surprising) selection of primary texts of the nineteenth century translated into English where necessary. Unfortunately the ‘translation’ includes modernizing expressions and notations, sometimes to such a degree that some passages are barely recognizable from the originals.

For better sensitivity to the issues exhibited by another eminent mathematician, see the lecture delivered by Weil A. in 1978 to an International Congress of Mathematicians: “History of mathematics — why and how”, in his Collected papers, iii (Heidelberg, 1979), 434–42.

The principal means that I advocate to be used in education is ‘history-satire’, where the mathematics of the past is used as a bank of results and methods for some modern educational purpose: The general historical record is respected (for example, changes in practice over decades), but the nuances of historiography are usually set aside (see my “Not from nowhere: History and philosophy behind mathematical education”, International journal of mathematical education in science and technology, iv (1973), 421–53). Similar techniques could obtain for education in other sciences.

The British Society for the History of Mathematics, mentioned in Section 1, was formed in 1971 (chiefly by Dubbey J. M. and Whitrow G. J. ), partly to reflect an increase in interest in the subject but especially because societies in neighbouring disciplines were so apathetic, or even hostile and contemptuous, to it that a separate organisation had to be formed.

See my op. cit. (ref. 20). The aftermath “discussion” involved one particular theorem (International journal of mathematical education in science and technology, vi (1975), 252–3).

See my (ed.), History in mathematics education: Proceedings of a workshop held at the University of Toronto, Canada, July-August 1983 (Paris, 1987).

In a further misfortune, History of Mathematics is both too historical and too mathematical for modern professionalized philosophers, despite the considerable influence that mathematics has borne upon philosophy (many of the great philosophers were mathematicians) and still brings to many of its modern concerns (especially in the (mis-)use of logical systems and theories). But there has been some increase in interest in the history of philosophy recently (a society for the subject has been formed in Britain), and in response to an evident gap in the literature I launched in 1980 the journal History and philosophy of logic, a topic which was too logical and philosophical for historians, and too historical for logicians and philosophers.

I know of cases where the mathematician pursued orthodox research in order to gain promotion and tenure, and then exhibited his genuine interest in history. How large is the community of mathematicians which cannot find space for historians? One indication is a recent Combined membership list of the American Mathematical Society, the Mathematical Association of America, and the Society for Industrial and Applied Mathematics (Providence, 1987), which covers not only the USA but (through joint memberships of other societies) a number of mathematicians from other countries. The book lists around 43,000 persons, with the majority of them based in institutions of higher education.

See my Convolutions in French mathematics, 1800–1840: From the calculus and mechanics to mathematical analysis and mathematical physics (3 vols, Basel and Berlin, DDR, 1990); cited hereafter as Convolutions. Crosland M. P. , The Society of Arcueil … (London, 1967) remains unmatched as a general survey of the institutions that were operating during the Imperial period; my “Grandes écoles, petite Université: Some puzzled remarks on higher education in mathematics in France, 1795–1840”, History of universities, vii (1988), 197–225 looks more myopically, but in more detail, at those associated with mathematics up to the 1830s. Dhombres J. G. , “Mathématisation et communauté scientifique française (1775–1825)”, Archives internationales d'histoire des sciences, xxvi (1986), 249–93, contains a variety of information on the French mathematical community of the time.

An excellent example of forgotten importance is the Ourcq canal project (1802–25), which not only ran a long canal up to Paris but also involved the digging of the city's port at La Villette, a pair of extra canals running from there to link with the Seine to north and to south, and a system of subsidiary water systems to improve the supply to Paris. It was the major civil engineering project of the period, and affected the everyday life of the capital. The mathematics of the water-flow of large bodies, and various related topics, gained fresh attention. The main director of the project wrote his own extensive survey and history ( Girard P. S. , Mémoires sur le canal de l'Ourcq … (2 vols and 2 vols atlas, Paris, 1831–45); since then nothing of significance has been done on it … (a brief account is included in my Convolutions, ch. 8).

See especially Bradley M. , “Gaspard-Clair-François-Marie Riche de Prony …” (CNAA (London) Ph.D., 1984), chs 2 and 3, of which a small portion was published as her “Civil engineering and social change: The early history of the Paris École des Ponts et Chaussées”, History of education, xiv (1985), 171–83.

See, for examples among many, Ponteil C. , Histoire de l'enseignement de France … (Paris, 1966); and Hulin-Jong N. , L'organisation de l'enseignement des sciences (Paris, 1989). There seems to be a tradition among French historians to confine their accounts to the Université system (whose origins are outlined in the next clause); it is high time that it stopped.

In accord with the situation described in Section 2, Zwerling C. fails to notice the relatively quick rise of mathematics in his excellent thesis on the history of science education at the École Normale (“The emergence of the École Normale Supérieure as a center of scientific education in nineteenth century France” (Harvard University Ph.D., 1976)). Prior to 1830 the École Normale could boast as its principal graduate in mathematics Cournot A. (who only studied there for one year). In 1831 it expelled Galois E. , whose name is one of the best-known in mathematics for his visualization of many essential features of group theory. One of the best-known stories in the history of mathematics is that Galois was killed in a duel in 1832, and teachers routinely enliven their teaching of group theory by retelling this event to their students. However, in line with mathematidans' normal view of history discussed in Section 3, they never think of building their teaching upon the action that Galois took the night before the duel — namely, to write down some exciting and profound mathematics. An historical book exists in which Galois's work is discussed and his essays are translated in English ( Edwards H. M. , Galois theory (Heidelberg, 1984)); but it is not normally recommended reading. Yet group theory (not only Galois's contributions to it) is a branch of mathematics marvellously suitable for teaching from an historical point of view — namely, its emergence from specific applications through a body of general results to axiomatized and uninterpreted forms instead of the unintelligible and unmotivated reverse order which is normally followed in teaching. On this history, see Wussing H. , Die Genesis des abstrakten Gruppenbegriffes (Berlin, DDR, 1969); English trans., The genesis of the abstract group concept (Cambridge, Mass., 1984).

See Index biographique de l'Académie des Sciences du 22 décembre 1666 au 1” octobre 1978 (Paris, 1979), 5–97.

An excellent recent social history of French cartography fails to mention Puissant even once, although he was the leading mathematical cartographer in France for the first half of the nineteenth century; unfortunately his work was full of sums ( Konwitz J. , Cartography in France 1660–1848 … (Chicago 1987)). Bret P. (Paris) is currently studying French cartography in depth.

See, for example, Kuhn T. S. , “Mathematical vs. [sic] experimental traditions in the development of physical science”, Journal of interdisciplinary history, vii (1976–77), 1–31; also in his The essential tension (Chicago and London, 1977), 31–65.

Among the MAP members, Fresel and Malus were probably the best experimenters; and Ampère and Fourier at least tried, Ampère's experiments being ingenious in conception. Biot was enthusiastic but sometimes very sloppy. In addition, CME member Hachette was unusual in working from time to time in electricity, and the collaborative work done by Lamé with his CME friend Clapeyron in Russia (on which see Bradley M. , “Franco-Russian engineering links: The careers of Lamé and Clapeyron, 1820–1830”, Annals of science, xxviii (1981), 291–312) shows that he could have become a major CME member had he wished to concentrate in those areas. But the division proposed fits the community as a whole remarkably well.

See my “Modes and manners of applied mathematics: The case of mechanics”, in Rowe and McCleary , op. cit. (ref. 9).

On the physical aspects of these developments, see Fox R. , “The rise and fall of Laplacian physics”, Historical studies in the physical sciences, iv (1974), 81–136; on the mathematical ones see my “From Laplacian physics to mathematical physics, 1805–1826”, in Burrichter C. Inhetveen R. and Kötter R. (eds), Zum Wandel des Naturverständnisses (Paderborn, 1987), 11–34, and also my Convolutions, ch. 7.

There may a partial case of this kind to be made for Fresnel: The origins of his waval conception of light are not clear. For a recent history of optics of this period which takes mathematics seriously, see Buchwald J. Z. , The rise of the wave theory of light … (Chicago, 1989).

See my “Work for the workers: Advances in engineering mechanics and instruction in France, 1800–1830”, Annals of science, xli (1984), 1–33; and for somewhat more detail my Convolutions, ch. 16.

The term “social physics” is often credited to Quetelet A. ; but Comte claimed priority for it (to 1822) in a typically sarcastic footnote against “a Belgian” ( Comte A. , Cours de philosophie positive, iv (Paris, 1839), 7).

See the documents transcribed in my “Recent researches in French mathematical physics of the early 19th century”, Annals of science, xxxvii (1981), 663–90, pp. 684–90, and in Convolutions as Document 20.8.

Fourier had produced much of his work on heat theory and Fourier analysis between 1804 and 1812, and it was known to Lagrange, Laplace, Biot and Poisson (mostly with negative consequences for Fourier); but it received publication and widespread attention only after 1816. See my Convolutions, chs 9 and 12.

See especially Ben-David J. , “The rise and decline of France as a scientific centre”, Minerva, viii (1970), 160–79; and Outram D. , “Politics and vocation: French science 1793–1830”, The British journal for the history of science, xiii (1980), 27–43.

H. Paul provides a welcome exception, although even he only mentions mathematics and does not discuss its development, at either the research or educational levels (“The issue of decline in nineteenth-century French science”, French historical studies, vii (1971–72), 416–51).